Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2006  June  Q#9
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Question
Given that f(x)=( x^{2} – 6x)(x – 2) + 3x,
a. express f(x) in the form x(ax^{2} + bx + c), where a, b and c are constants.
b. Hence factorise f(x) completely.
c. Sketch the graph of y = f(x), showing the coordinates of each point at which the graph meets the axes.
Solution
a.
We are given;
b.
c.
We are required to sketch
We have seen in (a) that it can written as;
It is a cubic function.
ü Find the sign of the coefficient of . This gives the shape of the graph at the extremities.
ü Find the point where the graph crosses yaxis by finding the value of when .
ü Find the point(s) where the graph crosses the xaxis by finding the value of when . If there is repeated root the graph will touch the xaxis.
ü Calculate the values of for some value of . The is particularly useful in determining the quadrant in which the graph might turn close to the yaxis.
ü Complete the sketch of the graph by joining the sections.
Sketch should show the main features of the graph and also, where possible, values where the graph intersects coordinate axes.
First we find xintercepts.
The point at which curve (or line) intercepts xaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
We have found in (b) that given cubic function can be written as;
Therefore;
Now we have three options.








Therefore, the graph intersects xaxis at three points (0,0), (3,0) and (5,0).
Next we find yintercept.
The point at which curve (or line) intercepts yaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Therefore;
Hence, the graph intercepts yaxis at (0,0).
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